To obtain the desired lower bound for the volume of \(C_n\) , it now remains
to obtain an appropriate upper bound for the volume of the region
\(D_n^{*}\) . The following lemma expresses this volume in terms of a
combinatorial quantity counting alternating permutations. Here a
permutation of a set of \(n\)  distinct real numbers \(x_1,\dots ,x_n\)  is
called up-down alternating if it satisfies \(x_1<x_2>x_3<\dots \) ,
i.e., if \(x_{i+1}-x_i\)  is positive for odd \(i\)  and negative for even \(i\) ;
see André [1]} for more about such permutations.
